Difference equations:: Detecting oscillations via higher order characteristic equations

In these notes we introduce an alternative procedure to detect the presence of a wide range of oscillatory solutions - the exponential oscillatory ones - in both continuous-time and discrete linear difference equations. The technique is addressed to the scalar case of the linear two-delay difference equation, but it can be extended to other types of evolution equations including the nonlinear ones, in higher dimensions. In particular, we give an illustration of its applicability to the case of a simple two dimensional difference equation, Oscillatory solutions are presented in their closed formulae.